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The notion of graded consequence has been explored further in a more generalized framework. The agreement between the syntactic and semantic notions of consequence is fully established. A notion of degree of consistency is introduced and some of its properties have been studied.

Summary. Rough set theory has seen nearly two decades of research on both foundations and on diverse applications. A substantial part of the work done on the theory has been devoted to the study of its algebraic aspects. 'Rough algebras' now abound, and have been shown to be instances of various algebraic structures, both well-established and relatively new, e.g., quasi-Boolean, Stone, double Stone, Nelson, Lukasiewicz algebras, on the one hand, and topological quasi-Boolean, prerough and rough algebras, on the other. More interestingly and importantly, some of these latter algebras find a new dimension (interpretation) through representations as rough structures. An attempt is made here to present the various relationships and to discuss the representation results.There have been other directions to the algebraic studies, too, especially in the development of relation and information algebras as well as rough substructures of algebraic structures, such as groups and semigroups, and structures on generalized approximation spaces and information systems. A brief overview of these approaches is also presented in this chapter.